On Gravity

Matter bends the space around it — but how much it bends depends on how that matter is distributed.

Consider a kitten on a mattress. It will make no visible indentation on the mattress.
Now consider 1000 kittens all arranged in a grid on the mattress: still no visible indentations.

Now move more kittens into the center of the mattress and, gradually, an indentation will form.
Furthermore, if we swap the mattress for actual spacetime, and add a dense enough region of kittens, the curve becomes so deep that light can't escape and we are left with a black hole! (And kitten spaghetti...)

$$\kappa$$

In Newtonian gravity the potential follows a strict inverse–square form. This assumes a uniform mass distribution. Real systems contain gradients, shear, and density variations that alter the effective curvature.

A small modification to the curvature term in the action,

introduces an exponential correction to the weak–field potential. The resulting effective potential takes the form

and the corresponding acceleration becomes

The parameter κ enters as the weak–field imprint of this geometric modification. It encodes how local structure modifies the effective curvature.

$$\kappa(r)$$

The geometric origin implies that κ depends on the local environment. Observationally, the dominant contributions arise from background curvature, velocity shear, and density.

κ increases in regions with strong shear or enhanced density, and decreases in smooth or diffuse environments. This produces the observed variation in gravitational behaviour across galaxies, clusters, and large–scale structure.

Geometric Algebra View (Optional)

For a deeper geometric intuition, consider κ in Clifford algebra Cl(1,3) — the language of spacetime rotations and oriented areas. In this picture, curvature generated by density gradients is represented by a bivector, an oriented plane element.

where:

This bivector B defines the plane in which radial paths react to structure. Its magnitude |B| measures how strongly matter clumping twists or redirects those paths. κ can be interpreted as an effective scalar built from |B|, encoding how local structure modifies the gravitational field.

This complements the f(R) derivation used above: exponential behaviour in modified gravity emerges naturally from geometric “wedges” in the Ricci curvature. Teleparallel analogues such as f(T) = T exp(βT) offer a torsion-based formulation where T ∼ |B|² links directly to the same bivector structure (Nojiri 2007; Farrugia 2016).

Vera Rubin stars

When astronomers calculated how fast stars should orbit in a galaxy, they used the standard intuition that stars near the center should orbit fast, and stars farther out should orbit much slower, because they are farther from most of the galaxy’s central mass. However Vera Rubin's observations contradicted this: the stars at the edges were not slowing down. They were moving just as fast as the stars near the center. In many galaxies, they move about three times faster than both Newton & Einstein predict.

Since κ adjusts gravity based on how matter is distributed, we can apply it directly to a real galaxy to see whether it reproduces the observed rotation speed:

To compare theory with rotation data, we derive κ directly from observations by taking the Newtonian speed from "baryonic" mass, compare to the observed speed, and solve for κ.

TLDR: considering density distribution seems to matter. (dark matter...)

Gravitational Lensing

The next question is whether this same curvature term applies to light as well as mass. Gravitational lensing allows us to test that directly by comparing the bending of light predicted from observed mass to the bending we actually observe.

In galaxy rotation, orbital velocity depends on the square root of the gravitational potential. This means the κ effect shows up as a factor of exp(κ·r / 2). In gravitational lensing, the bending of light depends on the potential directly, not its square root. So the same κ shows up as exp(κ·b / 2), where b is the light’s closest approach to the mass.

Collisions

During high-velocity cluster collisions, gas clouds experience shock compression and strong velocity shear, raising κ temporarily:

where

Gravitational lensing depends on the gravitational potential and increased κ multiplies the bending angle. As the shock and shear dissipate, κ_coll → 0 and the lensing map recenters naturally.

The lensing region shifts — appearing heavier — but "extra mass" is not needed when described as extra weight.

Bullet Cluster — Collision Shift

As the clusters pass through each other, the regions of strongest curvature shift — not because new mass appears, but because the collision briefly increases the weight of space itself.

Local Group — Basin Map

This map shows the gravitational potential of the Local Group as a continuous basin, where the Milky Way and Andromeda already share a merged gravity well explaining their future merger.

Supercluster Flow (2D)

The same gravitational potential equation can be applied to the large-scale mass distribution of our cosmic neighbourhood, yielding the shared basin of attraction that channels galaxies toward Virgo and the Laniakea core.

The flow arrows show the direction of gravitational infall (−∇Φ), illustrating how the Local Group is not isolated but part of a broader cosmic "supercluster" river system.

The same κ term used in galaxy rotation, lensing, and basin maps also enters the large-scale gravitational potential. When averaged over cosmological distances—dominated by voids rather than dense structures—it produces a small net positive contribution to the integrated potential: an emergent large-scale acceleration.

For large radii, expanding the exponential gives an effective acceleration

so that κ contributes a small outward term proportional to κ on large scales. When this contribution is averaged over the cosmic web, it acts in the same direction as a cosmological constant, but arises from structure rather than vacuum energy.

Effective acceleration term in the Friedmann equation

In a homogeneous background, the large-scale effect of κ can be summarised as an additional acceleration term in the Friedmann equation:

where 𝒜_κ is an effective contribution generated by the large-scale κ field. For a representative background value κ₀ ≈ 2.6×10⁻²⁶ m⁻¹ (from supercluster flows), the associated acceleration scale 𝒜_κ is of the same order of magnitude as the late-time acceleration usually attributed to Λ in ΛCDM.

In this view, the observed cosmic acceleration emerges from the cumulative effect of structure-dependent curvature, not from a fundamental vacuum energy term.

The Hubble Tension

The difference between early-universe and late-universe measurements of H₀ can be viewed through the same κ-lens as our supercluster flows. Local galaxies do not expand into empty space; they ride within coherent gravitational corridors shaped by κ-dependent structure.

Within these overdense regions, the effective expansion rate is slightly enhanced:

where β ≈ 1–2 parameterises how strongly local κ-dependent flows couple to the global expansion.

For a representative κ ≈ 8×10⁻⁴ Mpc⁻¹(corresponding to κ₀ ≈ 2.6×10⁻²⁶ m⁻¹) and r_local ≈ 100 Mpc with β ≈ 1.1:

H₀^(κ) ≈ 67 × (1 + 0.09) ≈ 73 km s⁻¹ Mpc⁻¹

This illustrates that the same κ–driven structural acceleration that shapes basin and supercluster flows can naturally generate a 5–10% enhancement in the locally inferred H₀, comparable to the Planck–SH₀ES tension.

Using the same gravitational potential, the acoustic angular scale of the CMB is:

θ_* ≈ 144.6 Mpc / 13.9 Gpc ≈ 0.0104 rad ≈ 0.60°
ℓ_* ≈ π / θ_* ≈ 301 (the first acoustic peak appears at ℓ ≈ 220 due to phase shift).

Because the intergalactic medium is extremely dilute, the density–weighted κ_eff along a typical line of sight is very small, so D_A — and hence θ_* — remains almost unchanged.

With a void–dominated line of sight:
κ_eff ≈ 3×10⁻²⁹ m⁻¹ and L ≈ 4.3×10²⁶ m → ½ κ_eff L ≈ 0.0065, so D_A^(κ) / D_A ≈ exp(0.0065) ≈ **1.0065** (≈ +0.65%).

Thus, the CMB acoustic scale remains intact, while κ contributes only a small, smooth, %–level correction to lensing.

Where sightlines intersect superclusters, this same factor enhances deflection slightly (typically 1–3%), consistent with the observed mild smoothing of the acoustic peaks.

Gravitational Waves in a κ–r Universe

Gravitational waves are one of our sharpest tests of gravity. In the κ–r geometry, present–day signals from neutron star and black hole mergers are indistinguishable from GR, while the same curvature response predicts enhanced primordial waves in the very early universe.

Today's detectors therefore see GR–exact waveforms, while the earliest gravitational waves are subtly reshaped by κ(r). The κ–r model passes current tests and makes falsifiable predictions for primordial signals.

Post-Newtonian Limit: GR Locally, κ₀ as a Small Correction

On Solar-System scales, any modification of gravity must reduce to the standard post-Newtonian form tested by planetary orbits and light deflection. In the κ–r framework this can be achieved by treating the κ-response as a very small correction to the time–time component of the metric.

With κ₀ at this level, the extra κ₀ r term in g_tt is completely negligible on Solar-System scales, so the standard post-Newtonian parameters remain indistinguishable from their GR values within current bounds, while still allowing a small cumulative effect from κ₀ on cosmological scales.

Mercury: the Famous 43″/Century Test

19th-century astronomers measured a tiny extra twist in Mercury’s orbit that Newtonian gravity couldn’t explain. General Relativity predicted an excess of about 43 arcseconds per century. The κ–r geometry matches the same result locally (with no extra parameters), and any cosmological bias from κ₀ is far below detectability.

The Pioneer Anomaly and the κ-Field

In the late 20th century, the Pioneer 10 and 11 spacecraft became the first human-made objects to leave the inner Solar System on long, clean, force-free trajectories. Their radio-tracking precision was unmatched: Doppler residuals were measured to parts in 10¹¹, far beyond what modern missions typically achieve. As the spacecraft passed beyond 10 AU, a persistent sunward acceleration appeared in the data:

Conventional analyses attribute this to thermal recoil from the RTGs. While plausible, this explanation requires fine-tuned directional emission and a nearly constant power asymmetry over decades. The anomaly remains unusually stable in magnitude despite the exponential decay of the plutonium heat source.

The magnitude of the anomaly aligns naturally with the expected background curvature scale:

This value is the same curvature amplitude that appears in galaxy rotation curves and weak-lensing fits across the κ-model. The Pioneer trajectory, extending from 1 AU to over 70 AU with minimal maneuvers, becomes a unique map of the Solar System’s κ-field.

κ₀ r × 10¹³ shows the tiny but cumulative lever arm of a background κ field across 20–70 AU. Even at the edge of the Pioneer range, κ₀ r ≪ 1, consistent with only a very small modification to Newtonian gravity in the outer Solar System.

Mass–Energy Equivalence in κ–Modified Gravity

The rest–energy relation E = mc² remains unchanged in the κ–model. Mass retains its inertial role. What changes is how energy couples to curvature. The effective gravitational mass acquires a scale–dependent weight through the factor exp(κ·r).

This introduces a distinction between inertial mass and gravitational mass without altering local special relativistic physics. At small radii, the exponential term approaches unity.

At galactic and cluster scales, the κ-term enhances gravitational interactions by weighting energy according to local density and shear. At quantum scales, the weighting disappears, and the conventional mass–energy equivalence governs the dynamics.

See Appendix A.6: “Mass–Energy Equivalence Under κ(r)”

Descent: The Quantum Limit

If κ encodes structure at every scale, where does that structure end?
What happens when r → ℓ_P — the quantum domain where mass and weight separate?

At Planck scales, κ loses leverage. Curvature decouples from structure.
The exponential vanishes, restoring the unweighted Newtonian (and GR) potential.

Energy gain vanishes at small r — but seeds the first structure at larger scales.

The transition defines a natural cutoff: below it, mass is inertial; above it, it carries geometric weight.
See PDF §3.8: "Quantum Scale Indications"

κ–Geometry and the Electromagnetic Coupling

The fine–structure constant α ≈ 1/137.036 is a dimensionless measure of electromagnetic interaction strength. Despite its importance, no accepted derivation from first principles exists. The κ–framework developed earlier introduces a curvature field built from local density, scale–dependence, and nonlinear response, suggesting a new geometric route for expressing α.

In this approach, electromagnetic coupling emerges from scale–weighted curvature. The same log–sensitive κ–structure used in the Riemann operator can be carried over to the hierarchy between the Bohr radius and the ultraviolet vacuum fluctuation scale. This produces a natural, dimensionless geometric quantity that can act as a coupling constant.

The goal is not to compute α numerically, but to identify a structural mechanism by which α could arise as a ratio of spectral quantities in a κ–geometry.

κ–Weighted Geometry Across Physical Scales

Electromagnetic interaction effectively interpolates between two characteristic lengths: the atomic scale where bound states form, and a short–distance scale where vacuum fluctuations dominate. Between these regimes the κ–curvature provides a natural log–weighted measure of how geometry changes with scale.

A κ–weighted geometric functional over a length scale ℓ is defined as

This quantity is dimensionless and encodes how curvature accumulates locally under log–uniform scaling. Objects of this type appear naturally in spectral geometry, where coupling constants arise from weighted ratios of geometric invariants extracted from an underlying operator.

Spectral Ratio Ansatz for the Fine–Structure Constant

The κ–operator introduced earlier,H = -d²/dt² + V(t), possesses a real spectrum and a well–defined heat kernel. These objects allow the construction of a dimensionless ratio analogous to those appearing in the spectral action expansion of Connes–Chamseddine theory. The proposal is that α arises from a κ–spectral ratio evaluated at a universal crossover scale ℓ₀.

Here ρₖ is the spectral density and Kₖ the heat–kernel amplitude of the κ– operator. A correct κ–geometry would produce a stable ratio close to 137, with its known slow running emerging from κ’s logarithmic scale dependence. The ansatz is structural rather than numerical, and provides a genuinely geometric starting point for understanding α.

Figure: κ–Weighted Geometry Across Scales

A schematic view of how the κ–curvature responds across physical scales, from atomic distances up to a universal crossover scale ℓ₀ and beyond. The idea is that electromagnetic coupling samples a narrow window of this curve.

Figure: Spectral Density and Heat Kernel in κ–Geometry

The κ–operator has a discrete spectrum with density ρκ(λ) and an associated heat kernel Kκ(ℓ). The fine–structure constant can be viewed as emerging from a ratio of these two quantities evaluated at a crossover scale ℓ₀.

Conceptual Summary: α as a κ–Spectral Ratio

In κ–geometry the electromagnetic coupling can be viewed as emerging from a balance between how many κ–modes are available at scale ℓ₀ and how strongly they are weighted by the heat kernel. The fine–structure constant is then interpreted as a compact way of encoding this balance.

In this picture, α is not a mysterious constant added by hand, but a compact descriptor of how κ–curvature modes are populated and weighted at a particular physical scale.

TOV Baseball: A Neutron Star in Your Hand

Imagine a fully loaded baseball diamond of neutron stars — four 1.4 M⊙ stars at the corners, 100,000 meters apart. Each packed with ρ ≈ 6.0 × 10¹⁷ kg/m³.

The central acceleration jumps from 0.85 m/s² to 0.99 m/s² — enough to trigger Schwarzschild collapse in under 1.5 km.

This shows how κ amplifies collapse in dense environments — the same mechanism that drives rapid SMBH formation in the early universe.

See PDF Section 3.4.1: "The TOV Baseball"

Supermassive Black Holes: Born Heavy

In dense, early-universe clouds, κ grows to 10⁻¹⁷ m⁻¹ — making gravity 16% stronger. Collapse accelerates. Accretion explodes. A 10⁹ M⊙ black hole forms in under 10 million years.

k-Curvature Operator

The same idea that curvature responds to local structure in gravity can be applied to the distribution of prime numbers. Instead of mass in space, we look at how primes are distributed along the integers and define a local “curvature” field built from nearby composites. For each integer n, let ρ(n) be the fraction of composite numbers in the window from n − 20 ton + 20. From this, we define a curvature coefficient

Passing to the continuous log–coordinate t = log x, we treat k_n as samples of a potential V(t)and define a Schrödinger–type operator acting on wavefunctionsψ(t):

Mathematically, H lives on a natural log–scale Hilbert space and, under mild conditions on V(t), is a self–adjoint operator with a real spectrum. The central idea is that the oscillations in V(t) encode the same structure that appears in the zeros of the Riemann zeta function.

Spectral Peaks recovered from the κ-Curvature Field

Taking a Fourier transform of the curvature sequence k_non a logarithmic grid recovers a set of sharp spectral peaks. These peaks align numerically with the first tens of imaginary parts of the non–trivial zeros of the zeta function to better than a percent, and their statistical spacing matches the random–matrix behaviour known from high–precision studies of the zeros.

FFT of κ-curvature field kₙ (computed for the first 100,000 primes).
Peaks coincide with the first 50 non–trivial zeta zeros to within 0.06% mean error. Prime data range: n = 2…1,299,709.

• Mean kₙ: 2.361
• Median kₙ: 2.409
• kₙ in twin gaps (gap = 2): 38.2% satisfy kₙ < 0.6
• Mean kₙ in large gaps (> 50): 2.573

This provides a concrete Hilbert–space operator whose spectrum appears empirically tied to the zeta zeros and can be developed further into a full Hilbert–Pólya–style framework.

What κ Cannot Do — and Why That Matters

The κ–curvature field was introduced as a local response to structure: density, shear, clustering and fragmentation. In the context of the integers this structure is external: the distribution of primes, composites and logarithmic scaling is already present before κ is ever defined. κ does not dictate the primes — it measures their curvature.

This distinction becomes important when κ is applied outside the setting that produced it. In some systems, such as the Collatz map or arbitrary discrete iterations, the “environment” is not independent. Any local density one computes is generated by the orbit itself, rather than reflecting an underlying landscape. In such cases κ ceases to be a diagnostic field and becomes circular: the orbit defines κ, and κ cannot constrain the orbit.

This is not a failure of the κ–model. It is evidence that κ is detecting areal external structure in the primes — and is not a universal magic function that solves every dynamical system. If κ “worked” on Collatz, it would be a sign that κ was too flexible. The fact that it does not transfer is an important control test: κ is sensitive to number–theoretic geometry, not arbitrary iteration rules.

In physical terms: curvature only makes sense when there is a geometry. κ succeeds on prime statistics for the same reason it succeeds on galaxies, clusters and the Pioneer trajectory: these systems possess a real underlying structure that curvature can measure. κ does not invent a landscape — it reveals one when it is already there.

This boundary is healthy. It shows that the κ–framework is grounded in structure, not numerology. The predictive success of κ in astrophysics and in the spectral analysis of the primes is meaningful precisely because κ does not apply everywhere. It applies where geometry is present.

κ–Geometry and the Electromagnetic Coupling

The fine–structure constant α ≈ 1/137.036 is a dimensionless measure of electromagnetic interaction strength. Despite its importance, no accepted derivation from first principles exists. The κ–framework developed earlier introduces a curvature field built from local density, scale–dependence, and nonlinear response, suggesting a new geometric route for expressing α.

In this approach, electromagnetic coupling emerges from scale–weighted curvature. The same log–sensitive κ–structure used in the Riemann operator can be carried over to the hierarchy between the Bohr radius and the ultraviolet vacuum fluctuation scale. This produces a natural, dimensionless geometric quantity that can act as a coupling constant.

The goal is not to compute α numerically, but to identify a structural mechanism by which α could arise as a ratio of spectral quantities in a κ–geometry.

κ–Weighted Geometry Across Physical Scales

Electromagnetic interaction effectively interpolates between two characteristic lengths: the atomic scale where bound states form, and a short–distance scale where vacuum fluctuations dominate. Between these regimes the κ–curvature provides a natural log–weighted measure of how geometry changes with scale.

A κ–weighted geometric functional over a length scale ℓ is defined as

This quantity is dimensionless and encodes how curvature accumulates locally under log–uniform scaling. Objects of this type appear naturally in spectral geometry, where coupling constants arise from weighted ratios of geometric invariants extracted from an underlying operator.

Spectral Ratio Ansatz for the Fine–Structure Constant

The κ–operator introduced earlier,H = -d²/dt² + V(t), possesses a real spectrum and a well–defined heat kernel. These objects allow the construction of a dimensionless ratio analogous to those appearing in the spectral action expansion of Connes–Chamseddine theory. The proposal is that α arises from a κ–spectral ratio evaluated at a universal crossover scale ℓ₀.

Here ρₖ is the spectral density and Kₖ the heat–kernel amplitude of the κ– operator. A correct κ–geometry would produce a stable ratio close to 137, with its known slow running emerging from κ’s logarithmic scale dependence. The ansatz is structural rather than numerical, and provides a genuinely geometric starting point for understanding α.

Figure: κ–Weighted Geometry Across Scales

A schematic view of how the κ–curvature responds across physical scales, from atomic distances up to a universal crossover scale ℓ₀ and beyond. The idea is that electromagnetic coupling samples a narrow window of this curve.

Figure: Spectral Density and Heat Kernel in κ–Geometry

The κ–operator has a discrete spectrum with density ρκ(λ) and an associated heat kernel Kκ(ℓ). The fine–structure constant can be viewed as emerging from a ratio of these two quantities evaluated at a crossover scale ℓ₀.

Conceptual Summary: α as a κ–Spectral Ratio

In κ–geometry the electromagnetic coupling can be viewed as emerging from a balance between how many κ–modes are available at scale ℓ₀ and how strongly they are weighted by the heat kernel. The fine–structure constant is then interpreted as a compact way of encoding this balance.

In this picture, α is not a mysterious constant added by hand, but a compact descriptor of how κ–curvature modes are populated and weighted at a particular physical scale.

Appendix: Key Derivations

This appendix outlines the main steps behind the κ–modified gravity equations used in the text. Each derivation is shown in a compact, weak–field form suitable for galaxies, clusters, and large–scale structure.

Full derivations, MCMC fits, and code at:

• github.com/hasjack/on-gravity
• On Gravity - October 11th 2025 (PDF)
• On Gravity - October 8th 2025 (PDF)